2 Note That The Following Four Subquestions Are Independent Of Each Other A 10 Points Consider The Following Three 1 (304.04 KiB) Viewed 26 times
2. Note that the following four subquestions are independent of each other. (a) [10 points] Consider the following three vectors in R3 : 2 -4 6 -2 2 ai = -8 12 4 2 > a3 = 8 -- () --(1)--() (3) Is ai E span{az, a3}? Is 0 E span{a2, az}, where 0 = ? Justify your answers. 0 (b) [8 points] Suppose {bı, b2} is a linearly independent set in R?. Show that {bi 7b2, 7b2 + bı} is also a linearly indepedent set. Is bị a linear combination of bı – 7b2 and 7b2 + bı ? Justify your answers. (c) [5 points] Consider the real vector space R4. Let C1 = (2,-1,-1,-1), C2 (1, 2, -1, -3), C3 = (-1,3,0, -2) - = = Determine whether c1, C2, and c3 are linearly dependent. Find the dimension and a basis for the subspace span {C1, C2, C3}. (d) [7 points] Suppose that W = {Y1, Y2, ..., Yn} is a basis for R”. Show that if M is an n xn invertible matrix, then R= {Myı, MY2, ..., Myn} is linearly independent.
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(3) Is ai E span{az, a3}? Is 0 E span{a2, az}, where 0 = ? Justify your answers. 0 (b) [8 points] Suppose {bı, b2} is a linearly independent set in R?. Show that {bi 7b2, 7b2 + bı} is also a linearly indepedent set. Is bị a linear combination of bı – 7b2 and 7b2 + bı ? Justify your answers. (c) [5 points] Consider the real vector space R4. Let C1 = (2,-1,-1,-1), C2 (1, 2, -1, -3), C3 = (-1,3,0, -2) - = = Determine whether c1, C2, and c3 are linearly dependent. Find the dimension and a basis for the subspace span {C1, C2, C3}. (d) [7 points] Suppose that W = {Y1, Y2, ..., Yn} is a basis for R”. Show that if M is an n xn invertible matrix, then R= {Myı, MY2, ..., Myn} is linearly independent.